Abstract:

Consider the countable semilattice consisting of the
recursively enumerable Turing degrees. Although is known to
be structurally rich, a major source of frustration is that no
specific, natural degrees in have been discovered, except the
bottom and top degrees, and . In order to overcome
this difficulty, we embed into a larger degree structure which
is better behaved. Namely, consider the countable distributive
lattice consisting of the weak degrees (a.k.a., Muchnik
degrees) of mass problems associated with nonempty subsets
of . It is known that contains a bottom degree
and a top degree and is structurally rich. Moreover,
contains many specific, natural degrees other than and
. In particular, we show that in one has
. Here is the
weak degree of the diagonally nonrecursive functions, and
is the weak degree of the -random reals. It is known that
can be characterized as the maximum weak degree of a
subset of of positive measure. We now show
that
can be characterized as the maximum weak
degree of a subset of whose Turing upward
closure is of positive measure. We exhibit a natural embedding of
into which is one-to-one, preserves the semilattice
structure of , carries to , and carries to
. Identifying with its image in , we show that all
of the degrees in except and are incomparable
with the specific degrees , , and
in .

Introduction

A principal object of study in recursion theory going back to the
seminal work of Turing [36] and Post [25] has been the
countable upper semilattice of recursively enumerable
Turing degrees, i.e., Turing degrees of recursively enumerable sets
of positive integers. See the monographs of Sacks [27],
Rogers [26], Soare [35], and Odifreddi
[22,23].

A major difficulty or obstacle in the study of has been the lack
of natural examples. Although it has long been known that is
infinite and structurally rich, to this day no specific, natural
examples of recursively enumerable Turing degrees are known, beyond
the two examples originally noted by Turing: the Turing
degree of the Halting Problem, and the Turing degree of
solvable problems. Furthermore, and are respectively
the top and bottom elements of . This paucity of examples in
is striking, because it is widely recognized that most other
branches of mathematics are motivated and nurtured by a rich stock of
examples. Clearly it ought to be of interest to somehow overcome this
deficiency in the study of .

In 1999 [30,31] we defined a degree
structure, here denoted , which is closely related to , but
superior to in at least two respects. First, exhibits
better structural behavior than , in the sense that is a
countable distributive lattice, while is not even a lattice.
Second and more importantly, there are plenty of specific, natural
degrees in which are intermediate between and , the
top and bottom elements of . Thus does not suffer from the
above-mentioned lack of examples, which plagues .

In more detail, let be the lattice of weak degrees
(a.k.a., Muchnik degrees) of mass problems given by
nonempty subsets of . In 1999
[30] we showed that among the intermediate degrees in
is the specific degree associated with the set of
-random reals. The concept of -randomness was already well
known from algorithmic information theory [19]. After
1999, we and other authors [2,3,4,5,32,33,34] continued
the study of , using priority arguments to prove structural
properties, just as for . In addition, we [33]
discovered families of specific, natural, intermediate degrees in
related to foundationally interesting topics such as reverse
mathematics, Gentzen-style proof theory, and computational complexity.
Some additional degrees of this kind are presented in Sections
3 and 4 below.

The purpose of the present paper is to further clarify the
relationship between the semilattice and the lattice .
Namely, we exhibit a specific, natural embedding
which is one-to-one, preserves the semilattice structure of , and
carries the top and bottom elements of to the top and bottom
elements of . See Theorem 5.5 below. By identifying
with its image in , we place the recursively enumerable
Turing degrees into a wider context, where natural intermediate
degrees occur. We view this as a step toward overcoming the
above-mentioned difficulties concerning .

At the same time, our embedding of into fails to solve the
long standing open problem of finding a specific, natural,
intermediate degree within itself. Indeed, we shall see below
(Theorem 5.6) that, regrettably, all of the intermediate
degrees belonging to the image of under our embedding are
incomparable with all of the known natural intermediate degrees in
.

Background: and

In this section we review some basic information concerning the
semilattice and the lattice .

Throughout this paper we shall use standard recursion-theoretic
notation from Rogers [26] and Soare [35]. For
special aspects of mass problems and sets, a convenient
reference is [33].

We write
for the set of natural numbers,
for the space of total functions from into
, and for the space of total functions from
into . We sometimes identify a set
with its characteristic function
given by if , if
. For
and
we write
to mean that the Turing machine with Gödel number
and oracle and input eventually halts with output . In the
absence of an oracle , we write . For
we consider recursive functionals given by
for some
and all and . A function
is said to be recursive if there exists
such that for all . A set
is said to be recursively enumerable if it
is the image of a recursive function, i.e.,
for some recursive
.

For
we write to mean that is
Turing reducible to , i.e.,
. The Turing degree of , denoted
, is the set of all such that , i.e.,
and . The set of all Turing degrees is
partially ordered by putting
if and only if
. Under this partial ordering, the bottom element of
is
is recursive. Within
, the least upper bound of and is
where
and
for all . A Turing degree is said
to be recursively enumerable if
where
is recursively enumerable. The set of all
recursively enumerable Turing degrees is denoted . It is easy to
see that is closed under the least upper bound operation
inherited from . The top and bottom elements of are
and respectively, where
is the Turing
degree of the Halting Problem,
.
Thus is a countable upper semilattice with a top and bottom
element.

Definition 2.1
Let be subsets of . We say that is
weakly reducible to , written , if for all there exists such that . The weak
degree of , written , is the set of all such
that , i.e., and . The set of
all weak degrees is partially ordered by putting
if and only if . The concept of
weak reducibility goes back to Muchnik [21] and has
sometimes been called Muchnik reducibility.

Theorem 2.2 is a complete distributive lattice.

Proof.The least upper bound of and in is
where

The greatest lower bound of and in is
. The bottom element of is

and is
recursive.

Note that if and only if
, where is the
Turing upward closure of ,

Thus the lattice of weak degrees, , is inversely isomorphic to
the lattice of subsets of which are upward closed
with respect to . It follows that is a complete
distributive lattice.

Remark 2.3
There is an obvious, natural embedding of into given by
. Here is the singleton set
whose only member is . This embedding is one-to-one, preserves
the partial ordering relation and least upper bound operation from
, and carries to . Compare this with our
embedding of into in Theorem 5.5 below.

Definition 2.4
A predicate
is said to be
recursive if

if , and
if .
A set
is said to be if there
exists a recursive predicate
such that
. A set
is said to be if there exists
a recursive predicate
such
that
.
Other levels of the arithmetical hierarchy are defined similarly.

Definition 2.5
A set
is said to be recursively
bounded if there exists a recursive function such that
for all .

Definition 2.6
For
, a recursive homeomorphism of
onto is a recursive functional mapping
one-to-one onto such that the inverse functional
is recursive. In this case we say that and are
recursively homeomorphic.

Theorem 2.7 is recursively bounded if and
only if is recursively homeomorphic to a set
.

Theorem 2.10 is a countable distributive lattice with a top and bottom
element, denoted and respectively.

Proof.If and are subsets of , then so are
and . Thus is closed under the least
upper bound and greatest lower bound operations inherited from
. Clearly is countable, because there are only countably
many subsets of . Clearly
is the bottom element of . Let
be the set of completions of Peano Arithmetic. Identifying
sentences with their Gödel numbers, we may view as a
subset of . By Scott
[29],
is the
top element of . See also [33, Section 6].

Remark 2.11
Just like the countable semilattice , the countable
distributive lattice is known to be structurally rich.
Binns/Simpson [2,5] have shown that every
countable distributive lattice is lattice embeddable in every
nontrivial initial segment of . Binns
[2,3] has obtained the analog of
the Sacks Splitting Theorem for [27]. Namely, for
all
in there exist
such
that
and
.
(The analog of the Sacks Density Theorem for
[28] remains as an open problem.) These structural
results for are proved by means of priority arguments. They
invite comparison with the older, known results for , which
were also proved by means of priority arguments.

We end this section by mentioning some technical notions and results
concerning trees and almost recursiveness.

A finite sequence of natural numbers
is called a
string of length . The set of all strings is denoted
. The set of strings of 's and 's is denoted
. If are strings of length
respectively, then the concatenation

is a string of length . We write
if
for some . If is a string of
length , then for all
we have
defined by
for
, for . We write
if
for some
.

A tree is a set
such that, for all
, . A path through
is an
such that
. The set of all paths through is denoted
. We sometimes identify a string with its Gödel
number
. A tree is said to be
recursive if
is recursive.

Theorem 2.12 is if and only if for
some recursive tree
.
is if and only if for some
recursive tree
.

Theorem 2.15If
is and nonempty, then there exists
such that is almost recursive.

Proof.This is a restatement of the Hyperimmune-Free Basis Theorem of
[15, Theorem 2.4]. See also [33, Theorem 4.19].

Some specific degrees in

In this section we identify and characterize some specific, natural
degrees in , and we investigate their degree-theoretic
properties.

Definition 3.1
Let be the ``fair coin'' probability measure on
given by
for all
. Let be a Turing oracle. A
point is said to be -random if there does
not exist a recursive sequence of
sets
, , such that
and
. If
the st Turing jump of , where is
recursive and , then is said to be -random.

Thus is -random if and only if is random in the
sense of Martin-Löf [20], and is -random if
and only if is random relative to the Halting Problem. For a
thorough treatment of randomness and -randomness, see Kautz
[16] or Downey/Hirschfeldt [8]. We write

Proof.It is is well known (see for instance [33, Theorem 8.3])
that is . Relativizing to we see that
is -random is
,
i.e., relative to . Putting , we see
that is relative to . From this it
follows easily that is
.

Lemma 3.3Let
be , and let
be nonempty . Then we can find a
nonempty set
such that
.

Proof.First use a Skolem function technique to reduce to the case where
is . Namely, fix a recursive predicate such that
, and
replace by the set of all
such that
holds.
Clearly the latter set is and . Assuming now
that is a subset of , let be a
recursive subtree of
such that is the set of
paths through . We may safely assume that, for all and the length of , . Let be a
recursive subtree of such that is the set of paths
through . Define to be the set of strings
of the form

where
,
, and
for all the length of . Thus
is a recursive subtree of
. Let
be the set of paths through .
(Compare the construction of in Jockusch/Soare
[14].) It is straightforward to verify that
. Note that is and recursively bounded. By Theorem
2.7 we can find a set
which is recursively homeomorphic to . This completes the proof.

Proof.By Lemma 3.4 there are sets
such that
and
. Thus
and
. Clearly
, hence
. Clearly has no recursive member, i.e.,
. By [33, Section 7] or [15, Corollary
5.4], the Turing upward closure of is of measure ,
i.e.,
for all
of positive
measure. In particular
, i.e.,
. Summarizing, we have now shown that
.

It remains to show that
, i.e.,
. By Lemma 3.2, let be a
nonempty subset of . By the Almost Recursive Basis
Theorem 2.15, let be almost recursive. By Kautz
[16, Theorem IV.2.4(iv)] or Dobrinen/Simpson
[7, Remark 2.8], there is no almost recursive .
In particular, there is no such that . Suppose
there were
such that . By Theorem 2.14
let
be a total recursive functional such
that . Put
. We have
, hence
, hence is nonempty. Since and
are , it follows by [33, Theorem 4.4] that
is . Hence, by
[33, Lemma 8.8], is of positive measure.
Since
, it follows that the Turing upward
closure of is of positive measure, but this is a
contradiction. We have now shown that, for a particular ,
there is no such that
. Thus
, and this completes the proof.

Remark 3.7
As an application of Theorem 3.6, we can find essentially
undecidable, finitely axiomatizable theories and in the
first-order predicate calculus, with the following properties: every
-random real computes a completion of ; every -random
real but not every -random real computes a completion of .
This follows from Theorem 3.6 plus the well known,
general relationship between subsets of and
finitely axiomatizable theories. See [32, Theorem 3.18 and Remark
3.19] and Peretyatkin [24].

Theorem 3.8We can characterize as the maximum weak degree of a
subset of of positive measure. We can
characterize
as the maximum weak degree of a
subset of whose Turing upward closure is of positive
measure.

Proof.The first statement is [33, Theorem 8.10]. For the
second statement, assume that is a subset of
whose Turing upward closure is of positive
measure. A computation in the style of Tarski and Kuratowski
(compare Rogers [26, Section 14.3]) shows that
is . Since is
of positive measure, let
be of
positive measure. By Kautz [16, Lemma II.1.4(ii)] or
Dobrinen/Simpson [7, Theorem 3.3], we may assume that
is relative to . Relativizing [33, Lemma
8.7] to , we have . Since
, it follows that
,
hence . Furthermore, since is a nonempty
subset of , we have
. We now see that
, i.e.,
. On the other
hand, by Theorem 3.6 let be a subset of
such that
. Note that
, hence is of positive
measure. We have now shown that
is the maximum
such that is of positive measure.
This completes the proof of our theorem.

Corollary 3.9Let be a nonempty subset of . Then
if and only if is of positive measure.

Proof.If then trivially
, hence
is of measure . Conversely, if is of
positive measure, then by Theorem 3.8 we have .

Corollary 3.10We can find a set
whose Turing upward
closure is of positive measure yet does not include
any set of positive measure.

Proof.By Theorem 3.6 let
be such
that
. By Theorem 3.8,
is of positive measure. If there were a set
of positive measure, then by Theorem
3.8 we would have
, contradicting Theorem
3.6.

Proof.The first statement is [33, Theorem 8.12, part 3]. For
the second statement, let
be nonempty
subsets of with
and
. Trivially
.
Moreover
, hence
is of positive measure if and only if at least
one of and is of positive measure. By
Corollary 3.9 this means that
if and only if
at
least one of and .

Definition 3.12
As in [33, Section 7], say that
is
separating if there is a pair of disjoint, recursively
enumerable sets
such that
where
separates . In particular,
is separating.

Some additional, specific degrees in

In this section we identify some additional specific, natural,
intermediate degrees in related to diagonal nonrecursiveness.

Definition 4.1
A function
is said to be diagonally
nonrecursive if
for all . We put
where

is diagonally nonrecursive.
The Turing degrees of diagonally nonrecursive functions have been
studied by Jockusch [12]. In
particular, a Turing degree contains a diagonally nonrecursive
function if and only if it contains a fixed point free function, if
and only if it contains an effectively immune set, if and only if it
contains an effectively biimmune set. Thus we see that the weak
degree is recursion-theoretically natural.

Proof.By Theorem 3.6 we have
.
Clearly has no recursive member, i.e.,
. By
Giusto/Simpson [11, Lemma 6.18], for all there
exists such that
. Thus we have
. It remains to show that
.
By Kumabe [18] there is a diagonally nonrecursive function
which is of minimal Turing degree. But if is
-random, then is not of minimal Turing degree, because the
functions and defined by are Turing
incomparable (see for instance van Lambalgen [37]). This
proves
. An alternative reference for the
conclusion
is Ambos-Spies et al [1, Theorems
1.4 and 2.1].

Definition 4.4
Let
be the set of recursively bounded functions.
Thus we have

Put
. For an extended discussion of
the recursion-theoretic naturalness of
and related weak
degrees, see [33, Section 10].

Proof.A Tarski/Kuratowski computation shows that
is
. Let
be as in the proof of Theorem
4.2. By Lemma 3.3 we can find a nonempty
set
such that
. Thus
. By Ambos-Spies
et al [1, Theorems 1.4, 1.8, 1.9] we have
, and the rest is from Theorem
4.3.

the set of functions which are diagonally nonrecursive relative to
. Put
and, for each ,

Clearly
is a subset of . Put
.

Remark 4.7
Trivially
and
for all .
The proofs of Theorems 4.2 and 4.3 show that
and
for all . By Kumabe
[18] we have
, and we conjecture that
for all . Thus in we apparently have

Note also that the sequence
can be
extended into the transfinite.

Remark 4.8
We conjecture that, for all , is incomparable with
. Note also that these 's are not to be confused
with the
's of Simpson [33, Example 10.14].
Indeed, we conjecture that, for all and ,
is incomparable with
.

Remark 4.9
We do not know of any specific, natural degrees in outside the
interval from to
, except and . On
the other hand, by Theorem 4.5 and Remarks 4.7
and 4.8, the interval from to
within appears to be remarkably rich in specific, natural
degrees.

Embedding into

In this section we exhibit a specific, natural embedding of the
countable upper semilattice into the countable distributive
lattice .

Definition 5.1
A singleton is a point
such
that the singleton set is .

is an upper semilattice homomorphism of the Turing degrees of
singletons into .

Proof.The first statement is the special case of Lemma 3.3
with and . For the second statement, note that for
any
and
we have
, and
implies
. In particular this
holds when and are singletons and .

Lemma 5.3We have an upper semilattice homomorphism of the Turing
degrees
into , given by (1).
Moreover,
and
.

Proof.The first statement is a special case of Lemma 5.2,
because
if and only if is (see
Kleene [17, Theorem XI, page 293]), which implies that
is a singleton. It is easy to see that
. To show that
, by Kleene
[17, Theorem 38*, pages 401-402] let
be
. Then
and
, hence
.

Lemma 5.4If
are Turing degrees
, and if
,
then
if and only if
. In
particular, the restriction of to is one-to-one.

Proof.Let
be such that
and
. We must show that if and only if
. The ``only if'' part is trivial.
For the ``if'' part, suppose
. In
particular,
. If , then
, hence
, hence
by
the Arslanov Completeness Criterion [12], hence
, a contradiction. Thus . This proves our lemma.

We now obtain our main result.

Theorem 5.5We have an embedding
given by (1).
The embedding is one-to-one, preserves the partial ordering
relation and the least upper bound operation from , carries
to , and carries to .

Remark 5.7
A set
is said to be -REA if
where is recursively enumerable
and, for each
, is recursively enumerable
relative to and . A Turing degree is said to be
-REA if it contains an -REA set. Note that any -REA set is
a singleton. Hence by Lemma 5.2 we have
for all -REA Turing degrees . Jockusch
et al [13, Theorem 5.1] have generalized the Arslanov
Completeness Criterion to -REA Turing degrees. In our terms,
their result says that if is an -REA Turing degree for
some , then
if and only if
, in which case
. Therefore, letting
denote the set of Turing degrees which are
and
-REA for some , we have as in Theorem 5.5
an embedding

which is one-to-one, preserves the partial ordering relation and the
least upper bound operation from
, and carries to
and to . Moreover, for all
other than
and , we have as in Theorem 5.6 that
is incomparable with
.

Remark 5.8
More generally, given
such that
, we have
an embedding

defined by
. The embedding
is one-to-one, preserves the partial ordering relation
and least upper bound operation from
, carries to
, and carries to . If we set
, we
recover the embedding
of Remark 5.7. If
we set
and restrict to , we recover the embedding
of Theorem 5.5.